Home » Blog » Compute some properties of compound interest rates

Compute some properties of compound interest rates

Consider a bank account which starts with P dollars and pays an interest rate r per year, compounding m times per year.

  1. Prove that the balance in the bank account at the end of n years is

        \[ P \left( 1 + \frac{r}{m} \right)^{mn}. \]

For fixed values of r and n, the balance at the end of n years as m \to +\infty is given by

    \[ \lim_{m \to +\infty} P \left( 1 + \frac{r}{m} \right)^{mn} = Pe^{rn}. \]

We say that money grows at the annual rate r with continuous compounding if the amount of money after t years is denoted by f(t) is given by

    \[ f(t) = f(0)e^{rt}. \]

Give an approximate length of time for the money in a bank account to double if r = .06 and compounds:

  1. continuously;
  2. four times per year.

Incomplete. Sorry, I’ll try to get back to this soon(ish).

One comment

  1. Tyler says:

    I’ll give this a shot:

    (a): There isn’t really much to prove, is there? This is just the formula for compound interest.

    (b): Solve for t: 2P = Pe^(rt), ln(2) = rt, r = 0.06, so t = ln(2)/0.06.

    (c): Solve for n: 2P = P*((1+0.06/4)^(4n)), 2 = ((1+0.06/4)^(4n)), log base 1.015 (2) = 4n,

    (log base 1.015 (2))/4 = n.

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):